Range Of Capillary Numbers Research Articles

Lateral migration of a capsule in a plane Poiseuille flow in a channel

Three-dimensional numerical simulation is presented on the motion of a deformable capsule undergoing large deformation in a plane Poiseuille flow in a channel at small inertia. The capsule is modeled as a liquid drop surrounded by an elastic membrane which follows neo-Hookean law. The numerical methodology is based on a mixed finite-difference/Fourier transform method for the flow solver and a front-tracking method for the deformable interface. The methodology can address large deformation of a capsule over a wide range of capsule-to-medium viscosity ratio. An extensive validation of the methodology is presented on capsule deformation in linear shear flow and compared with the boundary-element/integral simulations. Motion of a capsule in wall-bounded parabolic flow is simulated over an extended period of time to consider both transient and steady-state motion. Lateral migration of the capsule towards the centerline of the channel is observed. Results are presented over a range of capillary number, viscosity ratio, capsule-to-channel size ratio, and lateral location. After an initial transient phase during which the capsule deforms very quickly, the flow of the capsule is observed to be a quasi-steady process irrespective of capillary number ( Ca ) , capsule-to-channel size ratio ( a / H ) , and viscosity ratio ( λ ) . Migration velocity and capsule deformation are observed to increase with increasing Ca and a / H , but decrease with increasing λ , and increasing distance from the wall. Numerical results on the capsule migration are compared with the analytical results for liquid drops, and capsules with Hookean membrane which are valid in the limit of small deformation. Unlike the prediction for liquid drops, capsules are observed to migrate toward the centerline for 0.2 ⩽ λ ⩽ 5 range considered here. The migration velocity is observed to depend linearly on ( a / H ) 3 , in agreement with the small-deformation theory, but non-linearly on Ca and the distance from the wall, in violation of the theory. Using the present numerical results and the analytical results, we present a correlation that can reasonably predict migration velocity of a capsule for moderate values of a / H and Ca .

Crossover from fractal capillary fingering to compact flow: The effect of stable viscosity ratios

Using a standard pore-level model, which includes both viscous and capillary forces, we have studied the injection of a viscous, nonwetting fluid into a two-dimensional porous medium saturated with a less viscous, wetting fluid, i.e., drainage with favorable viscosity ratios, M> or =1 . We have observed a crossover from fractal capillary fingering to standard compact flow at a characteristic time, which decreases with increased capillary number and/or viscosity ratio. We have tested an earlier prediction for the dependence of this crossover upon viscosity ratio and capillary number using our data for a wide-but-physical range of capillary numbers and viscosity ratios. We find good agreement between the predicted behavior and our results from pore-level modeling. Furthermore, we show that this agreement is not affected by changes in the random distribution of pore throat radii or by changes in the coordination number, suggesting that the prediction is universal, i.e., valid for any porous medium structure, as expected from the general nature of the derivation of the prediction. Furthermore, this agreement indicates that the prediction correctly accounts for dependence of the flow upon capillary number and viscosity ratios, thereby enabling predictions for interfacial advance and width as well as saturation and fractional flow profiles. Also this agreement supports the validity of the general theoretical development lending credence to the three-dimensional predictions.

Coalescence of two equal-sized deformable drops in an axisymmetric flow

The coalescence of two equal-sized deformable drops in an axisymmetric flow is studied, using a boundary-integral method. An adaptive mesh refinement method is used to resolve the local small-scale dynamics in the gap and to retain a reasonable speed of computation. The thin film dynamics is successfully simulated, with sufficient stability and accuracy, up to a film thickness of O(10−4) times the undeformed drop radius, for a range of capillary numbers, Ca, from O(10−4–10−1) and viscosity ratios from O(0.1–10). The results are compared with experimental results from our earlier studies as well as the simple scaling theory for film drainage. The collisions for time-independent flow simulating head-on collisions in the experimental studies show two distinctively different regimes. At lower capillary numbers, the interfaces of the thin film between the colliding drops remain almost spherical up to the point of film rupture, and the dimensionless drainage time scales as tdG∼Ca. At higher capillary numbers, the film becomes dimpled at an early stage of the collision process, and the rate of the film drainage significantly slows down after the dimple is fully formed. In this case, the drainage time scales approximately as tdG∼Ca4∕3. The simulation, using a Hamaker constant with a fixed value calculated via Lifshitz theory, qualitatively agrees with the experimental results for the higher capillary numbers but not for the lower capillary numbers. The critical conditions for head-on collisions are also examined when the internal circulation within the drop, caused by the external flow, arrests the film drainage. Collisions in a time-dependent flow are also examined to simulate glancing collisions. Although the simulations predict many aspects of the experimental results, the results are quantitatively accurate, in comparison to the experimental data, only for the lowest viscosity ratio of 0.19. The interfaces of the thin film locally bulge outward when the drops are being pulled apart due to the suction pressure. This local deformation causes a local minimum in the film thickness. At the larger offsets, the coalescence angle continuously increases with Ca up to the separation angle (θ=55°–58°), for Ca<Cac. At smaller offsets, however, the local deformation for θ>45° cannot induce film rupture, even though coalescence is observed experimentally for the higher viscosity ratios.

Foam–wall friction: Effect of air volume fraction for tangentially immobile bubble surface

Wall-slip is an important phenomenon for the rheological behavior of various dispersions, including foams. In a previous study [N.D. Denkov et al., Coll. Surf. A 263 (2005) 129] we showed experimentally that the viscous stress between a foam and sliding solid wall is proportional to Ca 1/2 for a foam generated from potassium carboxylate surfactants at air volume fraction of Φ 3D = 90% ( Ca = μV 0/ σ is the capillary number). This experimental finding was explained by a theoretical model, whose main assumptions were: (1) the foam bubbles had tangentially immobile surface, and (2) the viscous friction occurred only in the dynamic wetting films formed between the bubbles and the solid wall. The current study is a continuation of our previous work, with the major aim to clarify theoretically the effect of air volume fraction, Φ, on the foam–wall friction (the variation of Φ is equivalent to varying the ratio of the film to Plateau border radii, R F/ R P). By using the lubrication approximation, we show with numerical calculations that for bubbles with tangentially immobile surfaces, more generally, the bubble–wall friction force could be represented as a superposition of two components: friction inside the wetting film, which is proportional to Ca 1/2 and friction in the transition zone film-meniscus, which is proportional to ( A 1 Ca 3/4 − A 2 Ca), where A 1,2 are numerical constants depending on Φ. The numerical calculations show also that the term describing the friction in the transition zone film-meniscus, is very well approximated by A 3 Ca 2/3 in the range of capillary numbers 10 −6 ≤ Ca ≤ 10 −3. An explicit expression is proposed, which describes rather well the numerical results for the friction force. The calculations show that the relative contribution of the friction inside the film dominates at Φ 3D ≥ ca. 90%. However, at lower air volume fractions (i.e., at smaller ratio R F/ R P) the friction in the transition zone film-meniscus could be significant and the effective power law index could have a value close to 2/3, as predicted in Bretherton's model [F.P. Bretherton, J. Fluid Mech. 10 (1961) 166]. The predictions of the current model are verified with literature data about the bubble–wall friction force, as well as with original experimental results about the thickness of the dynamic wetting films, formed between bubbles and moving solid wall. The obtained results could be useful in the analysis of the particle–wall friction in other systems, such as emulsions and dispersions of gel particles. The theoretical approach could be modified to describe various phenomena, such as motion of drops/bubbles in capillaries (including microfluidic applications) and hydrodynamic lubrication of deformable solid particles.

Experimental study of substrate roughness and surfactant effects on the Landau-Levich law

In this work we present an experimental study of deviations from the classical Landau-Levich law in the problem of dip coating. Among the examined causes leading to deviations are the nature of the liquid-gas and liquid-solid interfaces. The thickness of the coating film created by withdrawal of a plate from a bath was measured gravimetrically over a wide range of capillary numbers for both smooth and well-characterized rough substrates, and for clean and surfactant interface cases. In view of the dependence of the lifetime of a film on the type of liquid and substrate, and liquid-gas and liquid-solid interfaces, we characterized the range of measurability of the film thickness in the parameter space defined by the withdrawal capillary number, the surfactant concentration, and substrate roughness size. We then study experimentally the effect of a film thickening due to the presence of surfactants. Our recent theory based on a purely hydrodynamic role of the surface active substance suggests that there is a sorption-controlled coating regime in which Marangoni effects should lead to film thinning. However, our experiments conducted in this regime demonstrate film thickening, calling into question the conventional wisdom, which is that Marangoni stresses (as accounted by the conventional interfacial boundary conditions) lead to film thickening. Next we examine the effect of well-characterized substrate roughness on the coated film thickness, which also reveals its influence on wetting-related processes and an effective boundary condition at the wall. In particular, it is found that roughness results in a significant thickening of the film relative to that on a smooth substrate and a different power of capillary number than the classical Landau-Levich law.

Encapsulated Drop Breakup in Shear Flow

We investigate the deformation and breakup in shear flow of an encapsulated drop in which both the core and shell are Newtonian fluids. The equations of motion are solved numerically using a level set method to track interface motion. We consider the case of a drop stretched to a given length in constant shear and then allowed to relax. A range of morphologies is produced, and novel kinematics occur, due to the interaction of the core and outer interfaces. A phase diagram is presented to describe the morphologies produced over a range of capillary numbers and core interfacial tensions.

Brittle to Plastic Transition in the Dynamic Mechanical Behavior of Partially Saturated Granular Materials

The effect of liquid viscosity, surface tension and strain rate on the deformation behavior of partially saturated granular material was studied over a ten order of magnitude range of capillary number (the ratio of viscous to capillary forces). Glass spheres of average size 35 microns were used to make pellets of 35% porosity and 70% liquid saturation. As the capillary number increased, the failure mode changed from brittle cracking to ductile plastic flow. This change coincided with the transition from strain-rate independent flow stress to strain-rate dependent flow stress noted previously [Iveson, S. M., Beathe, J. A., and Page, N. W., 2002, “The Dynamic Strength of Partially Saturated Powder Compacts: The Effect of Liquid Properties,” Powder Technol., 127, pp. 149–161]. This change in failure mode is somewhat counter-intuitive, because it is the opposite of that observed for fully saturated slurries and pastes, which usually change from plastic to brittle with increasing strain rate. A model is proposed which predicts the functional dependence of flow stress on capillary number and also explains why the flow behavior changes. When capillary forces dominate, the material behaves like a dry powder: Strain occurs in localized shear planes resulting in brittle failure. However, when viscous forces dominate, the material behaves like a liquid: Shear strain becomes distributed over a finite shear zone, the size of which increases with strain rate. This results in less strain in each individual layer of material, which promotes plastic deformation without the formation of cracks. This model also explains why the power-law dependency of stress on strain rate was significantly less than the value of 1.0 that might have been expected given that the interstitial liquids used were Newtonian.

Microscopic and macroscopic structure of the precursor layer in spreading viscous drops.

We study the spreading of viscous nonvolatile liquids on smooth horizontal substrates using a phase-modulated interference microscope with sufficient dynamic range to enable the simultaneous measurement of both the inner ("microscopic") length scale and the outer ("macroscopic") flow scale in addition to the intermediate matching region. The resulting measurements of both the apparent contact angle and the lateral scale of the precursor "wetting" film agree quantitatively with theoretical predictions for a van der Waal's liquid over a wide range of capillary numbers.

Flow of a long bubble in a square capillary

The fraction of a Newtonian liquid deposited on the walls of a square capillary as a function of capillary number is theoretically determined by solving the differential equation that governs the motion of a quasi three-dimensional flow. A theoretical analysis of the equation of continuity and motion is formulated in terms of the stream function and thereby a fourth-order differential equation is obtained by eliminating the pressure. This equation is solved analytically to obtain an eigenfunction solution for θ→ π/2, where θ is the angle between the normal of the gas–liquid interface and the axial direction. The relationship between the Ca, λ a, λ b and kd with respect to the amount of liquid deposited on the walls of the square channel is obtained implicitly. The fractional coverage of a Newtonian fluid on the walls of the capillary computed from this analysis is compared with that of the experimental data. It was observed that the deviation between the theoretical analysis and the experimental study occurs at the entire range of capillary number. The solution is valid for an interface that is almost parallel to the walls of the square channel.

Gas and condensate relative permeability at near-critical conditions: capillary and Reynolds number dependence

Measurement of gas and condensate relative permeabilities is typically performed through steady state linear coreflood experiments. Experiments are normally designed to minimize the saturation variations throughout the core so that the relative permeabilities can be evaluated through direct application of Darcy's Law to each phase. This study addresses relative permeability prediction from linear corefloods for low permeability, near-critical gas–condensate systems at velocities typical of the near wellbore region. In this regime, relative permeabilities are capillary number dependent and non-Darcy terms are significant. This paper looks at combining both the nonlinear (in velocity) terms into a single “effective relative permeability” term. Synthetic “experimental” data were generated numerically through simulation (and not from laboratory experiments). The synthetic data considered a range of capillary numbers at both low and high Reynolds numbers. Simplified correlations were developed to represent both the gas and condensate effective relative permeabilities. Effective relative permeabilities were then history-matched from the synthetic experimental data through non-linear regression. History matching results indicate that the Reynolds number should be included explicitly in the gas effective relative permeability correlation. Two-phase injection experiments are easier to interpret than single-phase injection experiments.

Bubble Shapes and Orientations in Low Re Simple Shear Flow

We present measurements of shape and orientation of air bubbles in a viscous Newtonian fluid deformed by simple shear. The apparatus is a variation of the “parallel band” device developed by G. I. Taylor. Previous experimental studies on low viscosity ratio, low Reynolds number (Re⪡1) bubble deformation have focussed on either small or large deformations (mostly small deformation) and have only qualitatively examined the orientation of bubbles except for small deformations. Our data set spans both the theoretical small deformation and high deformation limits. With these data we confirm theoretical relationships and assess the range of capillary numbers (Ca) over which theoretical relationships for shape and orientation of bubbles are appropriate. We also examine the geometry of deformed bubbles as they relax to a spherical shape once shear stresses are removed. Our data indicate that for extremely small Reynolds numbers and viscosity ratios, the small deformation theoretical relationship first developed by Taylor, is a good approximation for Ca<0.5. The large deformation results for both shape and bubble orientation derived by Hinch and Acrivos agree with our data for Ca>1 and Ca>0.5, respectively.

Free surface Stokes flow over topography

We study two-dimensional viscous flow over topographical features under the action of an external body force. The Stokes equations are written as a set of harmonic and biharmonic equations for vorticity and stream function. A direct biharmonic boundary integral equation method is used to transform these equations to a pair of integral equations on the boundary of the domain. These equations are solved with a preassumed free surface profile to obtain values of the flow variables on the boundary. The location of the free boundary is then updated by considering the normal stress condition and using an iteration technique. We have studied the flow over steps and trenches of different depths and for a wide range of capillary numbers, Ca. Our computation shows that for small Ca, the free surface develops a ridge before the entrance to a step down and a depression region right before a step up. The magnitude and location of these features depend on the capillary number and the step depth. For large capillary numbers the free surface nearly follows the topography and the ridge and depression are found to be exponentially small in the capillary number. On the other hand, our results agree well with lubrication theory for small capillary numbers on the order of 10−2 or less, even for steep features.

Laboratory Study Investigating Emulsion Formation in the Near-Wellbore Region of a High Water-Cut Oil Well

Summary An experimental study was conducted to determine the morphology of oil/water mixtures produced into a high water-cut well. This information is important for determining the feasibility of downhole oil/water separation. Core-flow and microflow experiments were performed to simulate the two-phase flow process in the near-wellbore region of a high water-cut well. In the core-flow experiments the morphology of oil/water mixtures leaving different types of porous media was observed and the droplet-size distribution was determined for different fluid properties and flow parameters. In the microflow experiments the flow regimes and breakup mechanisms of the oil phase in the pore space were visualized and observed. The experimental results show that the oil/water mixture emerging from the porous medium depends strongly on its wettability. When the porous medium is oil-wet, the produced oil is mainly in the form of large drops that are formed at the porous end face. From water-wet porous media, small oil droplets are produced at high flow rates; their sizes are typically of the pore size's order and are formed inside the porous medium. A correlation between the droplet size and relevant fluid, and flow and porous-medium parameters has been developed, allowing for the assessment of circumstances in which downhole separation can be problematic. The experimental results have also shown that in water-wet porous media a transition from capillary-dominated to dispersed flow takes place in the capillary number range investigated (10-5&amp;lt;Nc&amp;lt;10-3).

Film Drainage between Colliding Drops at Constant Approach Velocity: Experiments and Modeling

Experiments and modeling of the drainage of the thin liquid film between two deformable spherical drops approaching each other at constant velocity in another liquid are being presented. Two numerical models based on the lubrication theory have been developed considering the cases of immobile or mobile drop interfaces. The absolute film thickness and the thinning rate have been measured using laser interferometry for a wide range of capillary numbers. In all studied cases, the model with immobile interfaces was found to give the best predictions of the experimental time evolution of the film thickness and radial expansion. These results made it possible to derive a typical time scale of the drainage process.

The motion of long bubbles through viscoelastic fluids in capillary tubes

The penetration of long gas bubble through a viscoelastic fluid in a capillary tube has been studied in order to investigate the influence of viscoelastic material properties on the hydrodynamic coating thickness and local flow kinematics. Experiments are conducted for three tailored ideal elastic (Boger) fluids, designed to exhibit similar steady shear properties but substantially different elastic material functions. This allows for the isolation of elastic and extensional material effects on the bubble penetration process. The shear and extensional rheology of the fluid is characterized using rotational and filament stretching rheometers (FSR). The fluids are designed such that the steady-state extensional viscosity measured by the FSR at a Deborah number (De) greater than 1 differs over three orders of magnitude (Trouton ratio = 103–106). The experiment set up to measure the hydrodynamic coating thickness is designed to provide accurate data over a wide range of capillary numbers (0.01 < Ca < 100). The results indicate that the coating thickness in this process increases with an increase in the extensionally thickening nature of the fluid. Experiments are also conducted using several different capillary tube diameters (0.1 < D < 1 cm), in order to compare responses at similar Ca but different flow De. Suitable scaling methods and nonlinear viscoelastic constitutive equations are explored to characterize the displacement process for polymeric fluids. Bubble tip shapes at different De are recorded using a CCD camera, and measured using an edge detection algorithm. The influence of the mixed flow field on the bubble tip shape is examined. Particle tracking velocimetry experiments are conducted to compare the influence of viscoelastic properties on the velocity field in the vicinity of the bubble tip. Local shear and extension rates are calculated in the vicinity of the bubble tip from the velocity data. The results provide quantitative information on the influence of elastic and extensional properties on the bubble penetration process in gas-assisted injection molding. The bubble shape and velocity field information provides a basis for evaluating the performance of constitutive equations in mixed flow.

Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure-driven flows

The dynamic behaviour and stability of a two-dimensional immiscible droplet subject to shear or pressure-driven flow between parallel plates is studied under conditions of negligible inertial and gravitational forces. The droplet is attached to the lower plate and forms two contact lines that are either fixed or mobile. The boundary-integral method is used to numerically determine the flow along and dynamics of the free surface. For surfactant-free interfaces with fixed contact lines, the deformation of the interface is determined for a range of capillary numbers, droplet to displacing fluid viscosity ratios, droplet sizes and flow type. It is shown that as the capillary number or viscosity ratio or size of the droplet increases, the deformation of the interface increases and above critical values of the capillary number no steady shape exists. For small droplets, and at low capillary numbers, shear and pressure-driven flows are shown to yield similar steady droplet shapes. The effect of surfactants is studied assuming a fixed amount of surfactant that is subject to convective–diffusive transport along the interface and no transport to or from the bulk fluids. Increasing the surface Péclet number, the ratio of convective to diffusive transport, leads to an accumulation of surfactant at the downstream end of the droplet and creates Marangoni stresses that immobilize the interface and reduce deformation. The no-slip boundary condition is then relaxed and an integral form of the Navier-slip model is used to examine the effects of allowing the droplet to slip along the solid surface in a pressure-driven flow. For contact angles less than or equal to 90°, a stable droplet spreads along the wall until a steady shape is reached, when the droplet translates across the wall at a constant velocity. The critical capillary number is larger for these droplets compared to those with pinned contact lines. For contact angles greater than 90°, the wetted area between a stable droplet and the wall decreases until a steady shape is reached. The critical capillary number for these droplets is less than that for pinned droplets. Above the critical capillary number the droplet completely detaches for a contact angle of 120°, or part of it is pinched off leaving behind a smaller attached droplet for contact angles less than or equal to 90°.

A general approach for modelling complex fluids: its application to concentrated emulsions under shear

We present a general Lagrangian–Eulerian approach for modelling complex fluid flows. Our method can track the microstructure evolution of complex fluids under flow in a natural physical way. It is especially useful for mesoscopic modelling of complex fluids in which the characteristic time and length scales are usually several orders of magnitude higher than those for simple fluids. The method is illustrated by studying hydrodynamically interacting two-dimensional concentrated emulsions under Couette flow. We have carried out the simulations of concentrated emulsions under shear over a range of capillary number (0.1≤Ca≤0.6) with a fixed area fraction φ=0.4 and viscosity ratio λ=1. The results show that concentrated emulsions exhibit the characteristics of viscoelastic fluids: strong shear thinning of shear stress and non-zero of first normal stress difference. Deformable drops can glide past each other with less resistance than in the case of hard suspensions and hence reduce shear stress. We have found that the dependence of the first normal stress difference on shear rate changes from quadratic to linear as the shear rate increases. In the regime of larger capillary numbers, highly inhomogeneous behaviour of drops and the dumbbell shape of drops can be clearly observed.

Transient Thermocapillary Migration of Deformable Bubbles

This paper utilizes a computational approach to address the effect of bubble deformation on the transient thermocapillary migration of bubbles in a microgravitational environment. We use a finite volume method with interface tracking capability on a moving unstructured mesh. We consider a range of Reynolds and Marangoni numbers from 1 to 100 and a range of capillary numbers from 0.001 to 0.1. The transient migration of bubbles in which bubble deformation is significant is described.

Pressure driven disturbances on a thin viscous film

Here we investigate the effect of a steady pressure disturbance moving at a constant velocity along the interface of a thin viscous film flowing down an inclined plane. Both steady and unsteady solutions are determined for the interface. We find that it is possible for the steady solutions to have surface waves, either preceding the disturbance or behind it, depending on the values of the Bond and capillary numbers. In addition, for fixed Bond number, disturbance amplitude, and constant film heights both far upstream and downstream, there exists a finite range of capillary numbers for which no steady solutions exist. Within this range of capillary numbers, the transient solution develops a shock like profile which adjusts the heights upstream and downstream of the disturbance and grows in width.

Numerical simulation of a concentrated emulsion in shear flow

A three-dimensional computer simulation of a concentrated emulsion in shear flow has been developed for low-Reynolds-number finite-capillary-number conditions. Numerical results have been obtained using an efficient boundary integral formulation with periodic boundary conditions and up to twelve drops in each periodically replicated unit cell. Calculations have been performed over a range of capillary numbers where drop deformation is significant up to the value where drop breakup is imminent. Results have been obtained for dispersed-phase volume fractions up to 30% and dispersed- to continuous-phase viscosity ratios in the range of 0 to 5. The results reveal a complex rheology with pronounced shear thinning and large normal stresses that is associated with an anisotropic microstructure that results from the alignment of deformed drops in the flow direction. The viscosity of an emulsion is only a moderately increasing function of the dispersed-phase volume fraction, in contrast to suspensions of rigid particles or undeformed drops. Unlike rigid particles, deformable drops do not form large clusters.